Definition 2.3 (see [1]}) 

Let \(p,q,r,\sigma _{1},\sigma _{2} \in (0,\infty ]\) , \(\alpha _{3} \ge 0\) , \(\alpha ^{1}:=\lbrace \alpha ^{1}_{k}\rbrace \) , \(\alpha ^{2}:=\lbrace \alpha ^{2}_{k}\rbrace \)  be sequences of
positive numbers. Assume that a weight sequence \(\lbrace t_{k}\rbrace  \in X^{\alpha _{3}}_{\alpha ,\sigma ,p}\) . By \(\widetilde{B}^{l}_{p,q,r}(\mathbb {R}^{n},\lbrace t_{k}\rbrace )\)  we denote the class of
functions \(f \in L^{\text{\rm loc}}_{r}(\mathbb {R}^{n})\)  with finite quasi-norm
\(\widetilde{B}^{l}_{p,q,r}(\mathbb {R}^{n},\lbrace t_{k}\rbrace ):= \\:=\biggl (\sum \limits _{k=0}^{\infty } \Vert t_{k} \delta ^{l}_{r}(\cdot +2^{-k}I^{n}) f|L_{p}(\mathbb {R}^{n})\Vert ^{q}\biggr )^{\frac{1}{q}}+\Vert t_{0}\Vert f|L_{r}(\cdot +I^{n})\Vert |L_{p}(\mathbb {R}^{n})\Vert \) 
